﻿Electromagnetic Lines and Tubes. 717 



and equations (12) become 





(15) 



These equations, one for each tube, are complementary to 

 the corresponding equations expressing the flux theorems, 

 and show that the tubes give a complete representation of 

 the electromagnetic field. Equations (9) show that the 

 tubes, by the variation of their cross-sections, determine 

 the variations of R, equations (15) show that by their 

 twists, they determine the variations of a, along each of 

 the four coordinate directions. It should be noted, however, 

 that while the variation of R given by each tube — x, I, y, z — 

 is that along the length of the tube, the variation of a given 

 by the twist is along a direction perpendicular to the tube, 

 but specially associated with it, — /, x, z, y. 



§ 5. An Electromagnetic Five- Vector characteristic of 

 the Field. 



The meaning of these results can be made clearer by 

 observing that the geometrical construction by which the 

 electromagnetic lines passing through a given point have 

 been obtained still leaves their directions in hyperspace 

 subject to a certain amount of arbitrariness. They are only 

 uniquely fixed when the arbitrary initial hyperplane is given 

 (cf. § 3). In fact the final directions of the lines can be got 

 from an arbitrary initial set of axes by four successive rota- 

 tions in the planes of xy } xz } yl } and zl, and rotations in the 

 planes xl and yz are not required to produce collinearity of 

 e and h. They may be made of course, but they do not 

 affect the collinearity. Thus in a hyperplane in which col- 

 linearity is observed at a given point, let the observer suppose 

 that he is in motion with a velocity v along the .I'-axis, 

 i. e. along the direction of E and H. The Lorentz transfor- 

 mation shows that this will affect in no way the magnitudes 

 of E and H at the point ; it follows, therefore, that the 

 observer has no means of ascertaining his velocity in this 

 direction, or of concluding that he has none, by observation 

 of the field at this single point. We have therefore no right 

 to assume that this velocity is zero, but in a general theory 

 should write it as an arbitrarily given quantity. The corre- 

 sponding transformation is equivalent to rotating the axes 

 through an arbitrary angle in the plane of xl, and thus to 



